t-clique ideal and t-independence ideal of a graph

ABSTRACT

In this paper, we introduce and study families of squarefree monomial ideals called clique ideals and independence ideals that can be associated to a finite graph. A family of clique ideals with linear resolutions has been characterized. Moreover, some families of graphs for which the quotient ring of their clique ideal is Cohen–Macaulay are introduced and some homological invariants of the clique ideal of a graph G, which is the complement of a path graph or a cycle graph, are obtained. Also some algebraic properties of the independence ideal of path graphs, cycle graphs and chordal graphs are studied.

KEYWORDS:

• Cohen–Macaulay
• linear resolution
• shellable
• t-clique ideal

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 13D02, 13F55
• Secondary 16E05

Acknowledgments

The author would like to thank Fahimeh Khosh–Ahang for useful discussions during the preparation of this paper.